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                                <h2>
                                    5.4 惯性制导原理
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                                <span class="article-info">
                                    2024-05-17, 911 words, 4 min read
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                            <div class="post_content markdown">
                                <p class="md_block">
                                    <span class="md_line md_line_start md_line_end">
                                        <h2 id="一-惯性制导">一、惯性制导</h2>
<h4 id="1-惯性导航和惯性制导的区别">1、惯性导航和惯性制导的区别</h4>
<ul>
<li>惯性导航：将敏感元件（陀螺仪和加速度计）测量的数据经过导航系统的解算，得到<strong>飞行器的位置、速度和姿态角</strong>。</li>
<li>惯性制导，直接使用敏感元件测量的数据计算关机方程和导引方程，<strong>不需要机载计算机进行位置、速度和姿态角解算</strong>。</li>
</ul>
<h4 id="2-采用摄动制导策略的原因">2、采用摄动制导策略的原因</h4>
<p>受到弹载计算机能力与数学工具的限制，只能在摄动假设下，采用基于标称轨迹的泰勒展开计算非线性函数</p>
<h4 id="3-摄动制导设计">3、摄动制导设计</h4>
<ol>
<li>根据任务需求，离线设计一条<strong>标称弹道</strong></li>
<li>在标称弹道附近对运动方程进行线性化，推导<strong>摄动方程</strong></li>
<li>基于摄动方程，推导<strong>关机方程</strong>和<strong>导引方程</strong>，使飞行器在标称弹道附近飞行，以满足终端的航程、倾角和横向位移等条件</li>
</ol>
<h2 id="二-外干扰补偿原理">二、外干扰补偿原理</h2>
<h2 id="三-捷联惯性制导的惯性方程">三、捷联惯性制导的惯性方程</h2>
<h2 id="四-敏感元件的冗余配置">四、敏感元件的冗余配置</h2>
<h4 id="1-提高敏感元件的精度和可靠性的方法">1、提高敏感元件的精度和可靠性的方法</h4>
<p>1、提高陀螺仪或加速度计等单个惯性器件的可靠性，以降低故障的发生率；（制造工艺难度大）<br>
2、增加一定数量的陀螺仪和加速度计（分为正交配置和非正交配置）</p>
<h4 id="2-冗余配置方案">2、冗余配置方案</h4>
<p>1、正交配置方案</p>
<ul>
<li>陀螺仪采用6表正交配置方案，每个方向2只陀螺仪。</li>
<li>加速度计采用9表冗余配置方案
<ul>
<li>𝑋通道配置4只加速度计，充分保证𝑋向加速度测量通道的稳定性，提升测量精度；</li>
<li>𝑌和𝑍通道分别配置2只加速度计；</li>
<li>第9只加速度计配置在𝑌平面与𝑌轴成45°夹角，主要用于对Ｙ、Ｚ向加速度输出进行故障检测</li>
</ul>
</li>
</ul>
<p>优点：更有利于提高某特定方向的测量精度，计算误差小，制造难度低，实时性好</p>
<p>2、非正交配置方案（十二表冗余惯导）</p>
<p>由<strong>六个惯性器件</strong>组成的冗余系统，其测量轴分别沿十二面体的<strong>六个平面的法线方向配置</strong>，精度分配比较平均。</p>
<ul>
<li>优点：提高了导航性能和故障检测能力。</li>
<li>缺点：需要对加速度计和陀螺仪的输出进行附加计算，增加了飞行控制计算机的计算量，且会引起新的计算误差</li>
</ul>
<h4 id="3-冗余配置方案评价">3、冗余配置方案评价</h4>
<p>1、评价方式</p>
<p>导航性能和可靠度</p>
<p>2、导航性能</p>
<ul>
<li>
<p>导航性能用来描述配置方案的测量误差</p>
</li>
<li>
<p>𝑛个单自由度元器件的捷联惯导系统，元器件的量测输入公式为 $m=HX+\eta $</p>
<ul>
<li>𝒎为𝑛维测量矢量，𝑯为𝑛 × 3维配置矩阵，𝑿为3维待求矢量，𝜼为𝑛维噪声矢量</li>
</ul>
</li>
<li>
<p>性能指标公式：<span class='katex-error' title='ParseError: KaTeX parse error: Undefined control sequence: \abs at position 9: F=\sqrt{\̲a̲b̲s̲{G}}=\sqrt{\abs…'>F=\sqrt{\abs{G}}=\sqrt{\abs{(H^TH)^{-1}}}</span></p>
<ul>
<li>𝑭取值越小，噪声𝜼产生误差的椭球体积就越小，导航性能就越好</li>
</ul>
</li>
<li>
<p>性能指标公式推理：线性最小方差</p>
<img src="http://cos.pansis.site/202405151608496.png/abc123" alt="image-20240515160811353" style="zoom:40%;" />
</li>
</ul>
<p>3、可靠度</p>
<ul>
<li>可靠度用来描述配置方案对元器件故障的容错能力</li>
<li>可靠度公式：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">R(t)=e^{-\lambda t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathdefault mtight">λ</span><span class="mord mathdefault mtight">t</span></span></span></span></span></span></span></span></span></span></span></span> （单个元器件发生故障的概率为𝜆）</li>
<li>配置方案的平均无故障时间：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>M</mi><mi>T</mi><mi>B</mi><mi>F</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">MTBF=\int_{0 }^{\infty } R(t)dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.215112em;vertical-align:-0.35582em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8592920000000001em;"><span style="top:-2.34418em;margin-left:-0.19445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579000000000002em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35582em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span></span>
<ul>
<li>MTBF值越大，说明配置方案的可靠度越高。</li>
</ul>
</li>
</ul>
<figure data-type="image" tabindex="1"><img src="http://cos.pansis.site/202405151612143.png/abc123" alt="image-20240515161241059" loading="lazy"></figure>
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